Valuing defaultable bonds: an excursion time approach

Recently there has been some interest in the credit risk literature in models which involve stopping times related to excursions. The classical Black-Scholes-Merton-Cox approach postulates that default may occur, either at or before maturity, when the firm's value process falls below a critical threshold. In the excursion approach the duration of default, the time period from the financial distress announcement through its resolution, is explicitly modeled. In this contribution, we provide a review of the literature on excursion time models of credit risk. Moreover, we examine the effects on credit spreads structure of different specifications of the event that triggers default.Credit risk, structural models, excursion approach, default threshold, default probability.

Simulation techniques for generalized Gaussian densities

This contribution deals with Monte Carlo simulation of generalized Gaussian random variables. Such a parametric family of distributions has been proposed in many applications in science to describe physical phenomena and in engineering, and it seems also useful in modeling economic and financial data. For values of the shape parameter a within a certain range, the distribution presents heavy tails. In particular, the cases a=1/3 and a=1/2 are considered. For such values of the shape parameter, different simulation methods are assessed.Generalized Gaussian density, heavy tails, transformations of rendom variables, Monte Carlo simulation, Lambert W function

An efficient binomial approach to the pricing of options on stocks with cash dividends

In this contribution, we consider options written on stocks which pay cash dividends. Dividend payments have an effect on the value of options: high dividends imply lower call premia and higher put premia. While exact solutions to problems of evaluating both European and American call options and European put options are available in the literature, for American-style put options early exercise may be optimal at any time prior to expiration even in the absence of dividends. In this case numerical techniques, such as lattice approaches, are required. Discrete dividends produce a shift in the tree; as a result, the tree is no longer reconnecting beyond any dividend date. Methods based on non-recombining trees give consistent results, but they are computationally expensive. We analyze binomial algorithms and performed some empirical experiments.Options on stocks, discrete dividends, binomial lattices

On the efficient application of the repeated Richardson extrapolation technique to option pricing

Richardson extrapolation (RE) is a commonly used technique in financial applications for accelerating the convergence of numerical methods. Particularly in option pricing, it is possible to refine the results of several approaches by applying RE, in order to avoid the difficulties of employing slowly converging schemes. But the effectiveness of such a technique is fully achieved when its repeated version (RRE) is applied. Nevertheless, its application in financial literature is pretty rare. This is probably due to the necessity to pay special attention to the numerical aspects of its implementation, such as the choice of both the sequence of the stepsizes and the order of the method. In this contribution, we consider several numerical schemes for the valuation of American options and investigate the possibility of an appropriate application of RRE. As a result, we find that, in the analyzed approaches in which the convergence is monotonic, RRE can be used as an effective tool for improving significantly the accuracy.Richardson extrapolation, repeated Richardson extrapolation, American options, randomization technique, flexible binomial method

Cumulative Prospect Theory portfolio selection

We introduce elements of Cumulative Prospect Theory into the portfolio selection problem and then compare stock portfolios selected under the behavioral approach with those selected according to classical approaches, such as Mean Variance and Mean Absolute Deviation ones. The mathematical programming problem associated to the behavioral portfolio selection is highly non-linear and non-differentiable; for these reasons it is solved using a Particle Swarm Optimization approach. An application to the STOXX Europe 600 equity market is performed.We introduce elements of Cumulative Prospect Theory into the portfolio selection problem and then compare stock portfolios selected under the behavioral approach with those selected according to classical approaches, such as Mean-Variance and Mean Absolute Deviation ones. The mathematical programming problem associated to the behavioral portfolio selection is highly non-linear and non-differentiable; for these reasons, it is solved using a Particle Swarm Optimization approach. An application to the STOXX Europe 600 equity market is performed

Insurance premium calculation under continuous cumulative prospect theory

We define a premium principle under the continuous cumulative prospect theory which extends the equivalent utility principle. In prospect theory risk attitude and loss aversion are shaped via a value function, whereas a transformation of objective probabilities, which is commonly referred as probability weighting, models probabilistic risk perception. In cumulative prospect theory, probabilities of individual outcomes are replaced by decision weights, which are differences in transformed, through the weighting function, countercumulative probabilities of gains and cumulative probabilities of losses, with outcomes ordered from worst to best. Empirical evidence suggests a typical inverse-S shaped function: decision makers tend to overweight small probabilities, and underweight medium and high probabilities; moreover, the probability weighting function is initially concave and then convex. We study some properties of the behavioral premium principle. We also assume an alternative framing of the outcomes; then we discuss several applications to the pricing of insurance contracts

Indici di volatilità

Gli indici di volatilità sono strumenti finanziari innovativi che hanno come scopo principale la misurazione della volatilità implicita dei mercati a breve e medio termine. Il più noto e utilizzato è l’indice americano VIX, che viene divulgato in tempo reale dal CBOE e stima la volatilità a 30 giorni del famoso indice azionario S&P 500. Per il suo calcolo si considerano solo i prezzi di mercato di opzioni call e put out-of-the-money. Il valore dell’indice, pertanto, non solo risulta indipendente da ogni tipo di modello che può essere assunto per descrivere la dinamica dell’attività sottostante, ma consente anche di isolare la volatilità attesa dagli altri fattori che influenzano il prezzo delle opzioni quali i dividendi, i tassi di interesse e il tempo che manca alla scadenza. Il calcolo del VIX è basato su un’approssimazione discreta del valore teorico dei contratti di tipo variance swap e, in quanto tale, è inficiato da diversi errori che comportano delle implicazioni negative su numerosi strumenti finanziari negoziati sia sui mercati ufficiali che sui mercati OTC.Volatility indexes are innovative financial instruments whose main purpose is measuring implied volatility of the markets in the short and medium term. The best known and used one is the American VIX, which is published in real time by the CBOE and estimates the 30 days volatility of the famous S&P 500 stock index. For its calculation, only the market prices of out-of-the-money call and put options are considered. The value of the index, therefore, not only is independent of any type of model that can be used to describe the dynamics of the underlying asset, but also it allows to isolate the volatility effect from other factors influencing the price of options, such as dividends, interest rates and the time to maturity. The VIX calculation is based on a discrete approximation of the theoretical value of the variance swap contracts; such an approximation is affected by several errors that implies negative effects on several financial instruments traded both on official and OTC markets

Probability weighting functions

Cumulative prospect theory (CPT) has been proposed as an alternative to expected utility theory to explain irregular behavior by economic agents. CPT comprises two key transformations: one of outcome values and the other of objective probabilities. Risk attitudes are derived from the shapes of these transformations as well as their interaction. The focus of this contribution is on the transformation of objective probability, which is commonly referred as probability weighting function. We review different families of weighting functions proposed in the literature and study their features

An efficient application of the repeated Richardson extrapolation technique to option pricing

In financial engineering one has frequently to deal with approximate results that are obtained by iterative methods or computational procedures depending on some parameter (e.g. the time-step). Often the convergence of numerical schemes is slow and this may be a serious problem to their use in practice. For this reason, acceleration techniques, such as Richardson extrapolation, have been studied and applied. In this contribution, we implement an efficient numerical method based on repeated Richardson extrapolation for the valuation of American options, paying particular attention to the choice of both the sequence of stepsizes and the order. In particular, we apply the method to the randomization approach proposed by Carr (1998), thus improving its accuracy by choosing a convenient sequence of stepsizes